nonlinear dynamics of longitudinal vehicle braking vsdia 04, budapest, hungary, november 8-10, 2004
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![Page 1: Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d7e5503460f94a61430/html5/thumbnails/1.jpg)
Nonlinear Dynamics of Longitudinal Vehicle Braking
VSDIA 04, Budapest, Hungary, November 8-10, 2004
![Page 2: Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d7e5503460f94a61430/html5/thumbnails/2.jpg)
Outline
• Single-Wheel Braking Model (SWBM)– Tractive Properties– Choice of Dynamic States– Key Features of the EOM– Slip Dynamics
• Two-Wheel Braking Model (2WBM)
• Closing Remarks
![Page 3: Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d7e5503460f94a61430/html5/thumbnails/3.jpg)
Single-Wheel Braking Model
• Governing Equations
![Page 4: Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d7e5503460f94a61430/html5/thumbnails/4.jpg)
Tractive Properties
s Longitudinal Wheel Slip
Longitudinal Friction Characteristic
Friction Law
![Page 5: Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d7e5503460f94a61430/html5/thumbnails/5.jpg)
Tractive Properties
• Dimensionless measure of the difference between u and R
Wheel Slip
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Tractive Properties
• Dimensionless measure of the difference between u and R
Wheel Slip
• In the steady-state:
1. Finite rotation R with s 2 , or
2. Lockup R with s
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Tractive PropertiesFriction Characteristics
“Magic Formula” of Pacejka
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Tractive PropertiesFriction Characteristics
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Choice of Dynamics States
• u and are coupled in a complicated way via the slip.
where
EOM in terms of u and
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Choice of Dynamics StatesEOM in terms of u and s
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Equations of MotionKey Features
• Slip dynamics are essentially decoupled
• Search for steady (constant) slip values s = s* that correspond to steady-state braking
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Slip Dynamics• Equilibria
• Local Stability
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State-Space Description
Stable Braking
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State-Space Description
Impending Possible Lockup
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State-Space Description
Possible Lockup
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State-Space Description
Impending Guaranteed Lockup
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State-Space Description
Guaranteed Lockup
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Bifurcation of Slip Dynamics
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Lockup Instability
• Critical Brake Torque
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Brake Torque Error
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Brake Torque Error
![Page 22: Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d7e5503460f94a61430/html5/thumbnails/22.jpg)
Brake Torque Error
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Two-Wheel Braking Model
• Governing Equations
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Tractive Properties
• Friction Laws
• Wheel Slip
• Friction Characteristic
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Equations of Motion• Same structure as EOM for SWBM
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Example State-Space Description
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Bifurcation of Slip Dynamics
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Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEELStable Braking
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Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEELImpending Possible Lockup
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Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEEL
Possible Lockup
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FRONT WHEELStable Braking
REAR WHEEL
Impending Guaranteed Lockup
Bifurcation of Slip Dynamics
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Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEEL
Guaranteed Lockup
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Lockup Instability
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Lockup Instability
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Closing Remarks
• Use of wheel slip s provides new insight into vehicle traction;
• Entire dynamics (stability and bifurcation) are captured by hbs (SWBM) and hbis (2WBM);
• New lockup threshold.
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Acknowledgments
• BUTE Dept. of Applied Mechanics• US-Hungarian Joint Fund for Technological
Development• MSU Dept. of Mechanical Engineering• MSU Institute for Global Engineering Education• National Science Foundation